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<H1><A NAME="SECTION03490000000000000000"></A><A NAME="secsvd"></A>
<BR>
Error Bounds for the Singular Value Decomposition
</H1>

<P>
The singular<A NAME="11684"></A> value decomposition (SVD) of a
real <B><I>m</I></B>-by-<B><I>n</I></B> matrix <B><I>A</I></B> is defined as follows. Let 
<!-- MATH
 $r = \min (m,n)$
 -->
<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img159.gif"
 ALT="$r = \min (m,n)$">.
The SVD of <B><I>A</I></B> is 
<!-- MATH
 $A=U \Sigma V^T$
 -->
<IMG
 WIDTH="92" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img157.gif"
 ALT="$A=U \Sigma V^T$">
(
<!-- MATH
 $A=U \Sigma V^H$
 -->
<IMG
 WIDTH="94" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img161.gif"
 ALT="$A=U \Sigma V^H$">
in the complex case),
where
<B><I>U</I></B> and <B><I>V</I></B> are orthogonal (unitary) matrices and

<!-- MATH
 $\Sigma = {\mbox {\rm diag}}( \sigma_1 , \ldots , \sigma_{r} )$
 -->
<IMG
 WIDTH="159" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img158.gif"
 ALT="$\Sigma = {\mbox {\rm diag}}( \sigma_1 , \ldots , \sigma_r )$">
is diagonal,
with 
<!-- MATH
 $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{r} \geq 0$
 -->
<IMG
 WIDTH="179" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img629.gif"
 ALT="$\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{r} \geq 0$">.
The <IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img36.gif"
 ALT="$\sigma _ i $">
are the <B>singular values</B> of <B><I>A</I></B> and the leading
<B><I>r</I></B> columns <B><I>u</I><SUB><I>i</I></SUB></B> of <B><I>U</I></B> and <B><I>v</I><SUB><I>i</I></SUB></B> of <B><I>V</I></B> the
<B>left and right singular vectors,</B> respectively.
The SVD of a general matrix is computed by xGESVD or xGESDD
<A NAME="11689"></A><A NAME="11690"></A><A NAME="11691"></A><A NAME="11692"></A>
<A NAME="11693"></A><A NAME="11694"></A><A NAME="11695"></A><A NAME="11696"></A>
(see subsection <A HREF="node29.html#subsecdriveeig">2.3.4</A>).

<P>
The approximate error
bounds<A NAME="footfnm 0"><SUP>4.10</SUP></A>for the computed singular values

<!-- MATH
 $\hat{\sigma}_1 \geq \cdots \geq \hat{\sigma}_{r}$
 -->
<IMG
 WIDTH="106" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img630.gif"
 ALT="$\hat{\sigma}_1 \geq \cdots \geq \hat{\sigma}_{r}$">
are
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
| \hat{\sigma}_i - \sigma_i | \leq {\tt SERRBD} \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="153" HEIGHT="31" BORDER="0"
 SRC="img631.gif"
 ALT="\begin{displaymath}
\vert \hat{\sigma}_i - \sigma_i \vert \leq {\tt SERRBD} \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The approximate error bounds for the computed singular vectors
<IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.gif"
 ALT="$\hat{v}_i$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img632.gif"
 ALT="$\hat{u}_i$">,
which bound the acute angles between the computed singular vectors and true
singular vectors <B><I>v</I><SUB><I>i</I></SUB></B> and <B><I>u</I><SUB><I>i</I></SUB></B>, are
<A NAME="11706"></A>
<A NAME="11707"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
 WIDTH="194" HEIGHT="58" BORDER="0"
 SRC="img633.gif"
 ALT="\begin{eqnarray*}
\theta ( \hat{v}_i , v_i ) &amp; \leq &amp; {\tt VERRBD}(i) \\
\theta ( \hat{u}_i , u_i ) &amp; \leq &amp; {\tt UERRBD}(i) \; \; .
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
These bounds can be computing by the following code fragment.
<A NAME="11714"></A>
<A NAME="11715"></A>

<P>
 
<P>
<PRE>
      EPSMCH = SLAMCH( 'E' )
*     Compute singular value decomposition of A
*     The singular values are returned in S
*     The left singular vectors are returned in U
*     The transposed right singular vectors are returned in VT
      CALL  SGESVD( 'S', 'S', M, N, A, LDA, S, U, LDU, VT, LDVT,
     $              WORK, LWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'SGESVD did not converge'
      ELSE IF ( MIN(M,N) .GT. 0 ) THEN
         SERRBD  = EPSMCH * S(1)
*        Compute reciprocal condition numbers for singular vectors
         CALL SDISNA( 'Left', M, N, S, RCONDU, INFO )
         CALL SDISNA( 'Right', M, N, S, RCONDV, INFO )
         DO 10 I = 1, MIN(M,N)
            VERRBD( I ) = EPSMCH*( S(1)/RCONDV( I ) )
            UERRBD( I ) = EPSMCH*( S(1)/RCONDU( I ) )
10       CONTINUE
      END IF
</PRE>

<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>,
if

<!-- MATH
 ${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img397.gif"
 ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = \left( \begin{array}{ccc} 4 & 3 & 5 \\2 & 5 & 8 \\3 & 6 & 10 \\4 & 5 & 11 \end{array} \right) \; ,
\end{displaymath}
 -->


<IMG
 WIDTH="158" HEIGHT="93" BORDER="0"
 SRC="img634.gif"
 ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 4 &amp; 3 &amp; 5 \\ 2 &amp; 5 &amp; 8 \\ 3 &amp; 6 &amp; 10 \\ 4 &amp; 5 &amp; 11 \end{array} \right) \; ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then the singular values, approximate error bounds, and true errors are given below.

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\hat{\sigma}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img635.gif"
 ALT="$\hat{\sigma}_i$"></TD>
<TD ALIGN="CENTER"><TT> SERRBD</TT></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $| \hat{\sigma}_i - \sigma_i |$
 -->
<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img636.gif"
 ALT="$\vert \hat{\sigma}_i - \sigma_i \vert$"></TD>
<TD ALIGN="CENTER"><TT> VERRBD</TT>(<B><I>i</I></B>)</TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $\theta ( \hat{v}_i , v_i )$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img568.gif"
 ALT="$\theta ( \hat{v}_i , v_i )$"></TD>
<TD ALIGN="CENTER"><TT> UERRBD</TT>(<B><I>i</I></B>)</TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $\theta ( \hat{u}_i , u_i )$
 -->
<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img637.gif"
 ALT="$\theta ( \hat{u}_i , u_i )$"></TD>
</TR>
<TR><TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">21.05</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.3 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img638.gif"
 ALT="$1.3 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.7 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img639.gif"
 ALT="$1.7 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $6.7 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img640.gif"
 ALT="$6.7 \cdot 10^{-8}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $8.1 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img641.gif"
 ALT="$8.1 \cdot 10^{-8}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $6.7 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img640.gif"
 ALT="$6.7 \cdot 10^{-8}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.5 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img642.gif"
 ALT="$1.5 \cdot 10^{-7}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER">2.370</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.3 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img638.gif"
 ALT="$1.3 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.8 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img643.gif"
 ALT="$5.8 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img644.gif"
 ALT="$1.0 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.9 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img645.gif"
 ALT="$2.9 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img644.gif"
 ALT="$1.0 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.4 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img646.gif"
 ALT="$2.4 \cdot 10^{-7}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER">1.143</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.3 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img638.gif"
 ALT="$1.3 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.2 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img536.gif"
 ALT="$3.2 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img644.gif"
 ALT="$1.0 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.0 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img647.gif"
 ALT="$3.0 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.1 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img648.gif"
 ALT="$1.1 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.4 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img646.gif"
 ALT="$2.4 \cdot 10^{-7}$"></TD>
</TR>
</TABLE>
</DIV>

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<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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